Group-Theoretic Partial Matrix Multiplication

TL;DR

This paper extends group-theoretic matrix multiplication algorithms to partial matrices, potentially improving bounds on matrix multiplication exponents and introducing new NP-hard optimization problems with algorithms.

Contribution

It introduces a novel partial matrix multiplication framework based on group theory, enhancing the flexibility and potential efficiency of matrix multiplication algorithms.

Findings

Partial matrix multiplication can improve upper bounds on matrix multiplication exponents.

Maximizing the 'fullness' of partial matrix patterns is NP-hard.

Provided algorithms for optimizing the partial matrix pattern.

Abstract

A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve upper bounds on the exponent of matrix multiplication yielded by group-theoretic full matrix multiplication. The group theory behind our partial matrix multiplication algorithms leads to the problem of maximizing a quantity representing the "fullness" of a given partial matrix pattern. This problem is shown to be NP-hard, and two algorithms, one optimal and another non-optimal but polynomial-time, are given for solving it.